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Initiation stage of nanosecond breakdown in liquid
View the table of contents for this issue, or go to the journal homepage for more
2014 J. Phys. D: Appl. Phys. 47 025502
(http://iopscience.iop.org/0022-3727/47/2/025502)
Home Search Collections Journals About Contact us My IOPscience
J. Phys. D: Appl. Phys. 47 (2014) 025502 (5pp) doi:10.1088/0022-3727/47/2/025502
Initiation stage of nanosecond breakdownin liquid
Mikhail Pekker1, Yohan Seepersad1,2, Mikhail N Shneider3,Alexander Fridman1,4 and Danil Dobrynin1
1 A J Drexel Plasma Institute, Camden, NJ, 08103, USA2 Department of Electrical and Computer Engineering, Drexel University, Philadelphia, PA, 19104, USA3 Department of Mechanical and Aerospace Engineering, Princeton University, Princeton,NJ, 08544, USA4 Department of Mechanics and Mechanical Engineering, Drexel University, Philadelphia,PA, 19104, USA
Received 23 September 2013, revised 28 October 2013Accepted for publication 4 November 2013Published 11 December 2013
AbstractIn this paper, based on a theoretical model (Shneider and Pekker 2013 Phys. Rev. E87 043004), it has been shown experimentally that the initial stage of development of ananosecond breakdown in liquids is associated with the appearance of discontinuities in theliquid (cavitation) under the influence of electrostriction forces. Comparison of experimentallymeasured area dimensions and its temporal development were found to be in a good agreementwith the theoretical calculations. This work is a continuation of the experimental andtheoretical works (Dobrynin et al 2013 J. Phys. D: Appl. Phys. 46 105201, Starikovskiy 2013Plasma Sources Sci. Technol. 22 012001, Seepersad et al 2013 J. Phys. D: Appl. Phys.46 162001, Marinov et al 2013 Plasma Sources Sci. Technol. 22 042001, Seepersad et al 2013J. Phys. D: Appl. Phys. 46 3555201), initiated by the work in (Shneider et al 2012 IEEE Trans.Dielectr. Electr. Insul. 19 1597–82), in which the electrostriction mechanism of breakdownwas proposed.
Keywords: Mikhail Pekker, Yohan Seepersad, Mikhail Shneider, Alexander Fridman, DanilDobrynin
(Some figures may appear in colour only in the online journal)
1. Introduction
The experimental study of nanosecond breakdown in the liquidphase has recently become possible with the developmentof stable and reliable high voltage nanosecond pulsed powersources [8]. Typically in the vicinity of the electrode, bubblesare generated due to ohmic heating via a standard breakdownmechanism [9]; however, this mechanism does not work in afew nanoseconds. Other mechanisms proposed in [10, 11] alsocould not explain the breakdown with pulses of such a shorttime.
The breakdown mechanism in water, based on theelectrostriction effect, was proposed in [2]. It is well knownthat in a non-uniform electric field, volumetric ponderomotiveforces pull a dielectric into the region with strongest field
[12, 13]. However, if the voltage rise time is very steep,the fluid does not have enough time to come into motiondue to inertia. Consequently, the ponderomotive forces causesignificant electrostrictive tensile stress in the dielectric liquid,which can lead to a disruption of the continuity of liquid(creating nanopores), similar to those observed in [14–16] andmany other experiments.
For the formation of discontinuities, it is necessary thatthe negative pressure in the fluid reach values on the orderof 10–30 MPa [14]. In this case, density fluctuations, whichalways exist due to thermal motion, lead to the formation ofgrowing cavitation bubbles (voids). However, in the case of along rise time of the high voltage pulse, ponderomotive forcesare able to cause movement and arising hydrostatic pressurecan compensate the negative electrostrictive pressure. In other
J. Phys. D: Appl. Phys. 47 (2014) 025502 M Pekker et al
Figure 1. Optical observation of the cavitation region. The radius of curvature of the electrode is R0 = 35 µm. The distance between theneedle tip and the flat electrode is dp = 1.5 mm. (a) The voltage between the electrodes is zero. (b) The voltage between the electrodes ismaximum.
words the sum of the electrostrictive and hydrostatic pressurescannot reach the threshold for the formation of growingcavitation bubbles. In [1] it is shown how quickly the voltageon the tip of electrode should grow, so that the total pressurenear the electrode reaches a critical value. The numericalmodel [1] was also used for modelling the experiments [5].
As it was noted in [2], voids in the liquid dielectric can bea source of secondary electrons. This is a necessary conditionfor the development of the discharge. If the cavities are largeenough, the electrons in them can gain sufficient energy toionize the molecules of water on the opposite side of thecavity. If the tip electrode is positively charged, the electronsquickly become neutralized on the electrode, and the remainingpositive charge becomes a non-uniform electric field source,which leads to the creation of new voids.
In the present work, on the basis of comparingexperimental results with numerical calculations, it was shownthat electrostriction forces really lead to creating nanopores,which determine the discharge formation in [3–7].
2. Numerical model
According to [1], the system of time-varying equations fora compressible fluid (water) in a strong electric field has theform:∂ρ
∂t+ ∇ (ρ �u) = 0 (1)
ρ
(∂ �u∂t
+ (�u · ∇) �u)
= −∇p +ε0
2
(∂ε
∂ρρ
)∇E2 (2)
p = (p0 + B)
(ρ
ρ0
)γ
− B (3)
ρ0 = 1000 kg m−3, p0 = 105Pa,
B = 3.07 × 108 Pa, γ = 7.5.
Figure 2. Time dependence of voltage on the electrode.
Here, ρ is the density, �u is the velocity, p is the hydrostaticpressure. Equation (1) is the continuity equation, (2) is themomentum equation, and (3) is the Tait equation of state, whichrelates the pressure to the density of water [17, 18]. The secondterm in the right side of (2) corresponds to electrostrictiveforces in a non-uniform electric field associated with thetensions within the dielectric, ε0 is the vacuum dielectricpermittivity, ε is the relative dielectric constant of the medium.In polar dielectrics: ∂ε
∂ρρ = αε, for water α ≈ 1.5 [9, 19].
In equation (2) friction is neglected because a few tens ofnanoseconds is not enough time for boundary layer formation(estimations are shown in [1]).
Boundary conditions: We assumed the no-slip condition (thefluid velocity at the electrode goes to zero) and the continuityof the fluxes of the density and momentum on both the
2
J. Phys. D: Appl. Phys. 47 (2014) 025502 M Pekker et al
9.72kV
12.6kV
20.0kV
Figure 3. Results of measurement. 0 ns corresponds to camera exposure from time t0 = 0 ns until t1 = 1 ns. 1 ns corresponds to cameraexposure from t0 = 1 ns until t1 = 2 ns. 2 ns corresponds to camera exposure from t0 = 2 ns until t1 = 3 ns. 3 ns corresponds to cameraexposure from t0 = 3 ns until t1 = 4 ns. 4 ns corresponds to camera exposure from t0 = 4 ns until t1 = 5 ns.
l=93µm20.0kV
Figure 4. The results of measurements for amplitude of 20 kV. 20 ns corresponds to camera exposure from time t0 = 20 ns until t1 = 21 ns.24 ns corresponds to camera exposure from t0 = 24 ns until t1 = 25 ns. 28 ns corresponds to camera exposure from t0 = 28 ns untilt1 = 29 ns. 38 ns corresponds to camera exposure from t0 = 38 ns until t1 = 39 ns. 88 ns corresponds to camera exposure from t0 = 88 nsuntil t1 = 89 ns.
electrode and on the boundaries of the computational domain.The system (1)–(3) was numerically calculated in prolatespheroidal coordinates [1, 20].
Since ε02 ( ∂ε
∂ρρ) is constant, we can rewrite the right part of
(2) as
−∇p +ε0
2
(∂ε
∂ρρ
)∇E2 = −∇
(p − 1
2αε0εE
2
)
= −∇ptotal. (4)
This means that the total pressure ptotal = p − 12αε0εE
2
acting on the dielectric liquid is a sum of hydrostatic p andelectrostriction-related pE = − 1
2αε0εE2 pressures.
3. Experimental setup
In this paper, we omit the details of the experiment, sincethey are the same as in [5, 7], and focus only on the idea ofthe method of detection of voids proposed in [2]. Since thesharp edge of the region with voids leads to the scatteringof light (the opalescence [21]), subtracting the illuminationon the screen with the voltage applied (figure 1(b)) by theillumination on the screen with no voltage (figure 1(a)), wecan easily define the cavitation boundary. The Schlierenmethod, which is a modification of the shadow [22], was usedin [5, 7]. In figure 2, the profiles of the voltage supplied tothe electrode are presented [7]. Figures 3 and 4 present theSchlieren images (image size 340 × 230 µm2). The ‘dark’area in the vicinity of the electrode shows cavitation. It can beobserved that there is less cavitation for a voltage V0 = 12.6 kVthan at V0 = 19.96 kV and at V0 = 9.72 kV cavitation is
totally absent. In all of our measurements, we did not observeemission in the vicinity of the electrode. It means that in allof our measurements we were always below the threshold ofbreakdown.
4. Numerical simulation
The results of calculations fulfilled on the basis ofequations (1)–(3) for the parameters similar to the experimentdescribed in this paper are presented in figure 5. The radiusof curvature of the needle electrode was R0 = 35 µm, theamplitude of the voltage on the electrode was V0 = 20, 12.2,10.0 kV, the time dependence of the voltage was varied inaccordance with figure 2: in time interval 0–4 ns, voltage grewlinearly, in 4–14 ns was constant, in 14–18 ns dropped linearly,in 18–150 ns, the voltage was zero.
It can be seen that due to the inertia, velocity of fluidflows are small, and does not exceed 1–5 m s−1 (third columnin figure 5). This means that in the experiments describedabove, the liquid does not have time to reach the electrode andcompensate the negative pressure related to the electrostrictiveforces p � |pE|, (see the first column 1 in figure 5). Thus,we can conclude that at the stage of growth amplitude of thevoltage, the refractive index is small and can be neglected (seefirst column on figure 5). The corresponding estimates confirmthis statement. The ‘dark’ region observed in figure 3 indicatesthe presence of voids (opalescence effect [21]). It should benoted that a threshold exists as illustrated by the absence of‘dark’ area that occurs at voltages less than 10 keV.
3
J. Phys. D: Appl. Phys. 47 (2014) 025502 M Pekker et al
0 20 40 60 80 100 120
-12
-10
-8
-6
-4
-2
0
2 1
2P[M
Pa]
z-zel [µm]
3
(a)
0 20 40 60 80 100 120-11
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
P[M
Pa]
z-zel [µm]
4
0 20 40 60 80 100 120-1.1
-1.0
-0.9
-0.8
-0.7
-0.6
-0.5
-0.4
-0.3
-0.2
-0.1
0.01
2
3
u[m
/s]
z-zel [µm]
4
0 20 40 60 80 100 120-20
-18
-16
-14
-12
-10
-8
-6
-4
-2
0
2
4
1
2P[M
Pa]
3
(b)
(c)
0 20 40 60 80 100 120-16
-14
-12
-10
-8
-6
-4
-2
0
1
2
3
P[M
Pa]
4
0 20 40 60 80 100 120-1.6
-1.4
-1.2
-1.0
-0.8
-0.6
-0.4
-0.2
0.01
2
3
u[m
/s]
z-zel [µm]z-zel [µm]z-zel [µm]
4
0 20 40 60 80 100 120
-50
-40
-30
-20
-10
0
101
2
P[M
Pa]
z-zel[µm]
3
0 20 40 60 80 100 120-45
-40
-35
-30
-25
-20
-15
-10
-5
0
1
2
3
P[M
Pa]
z-zel[µm]
4
0 20 40 60 80 100 120-4.4
-4.0
-3.6
-3.2
-2.8
-2.4
-2.0
-1.6
-1.2
-0.8
-0.4
0.0
3
4
1
u[m
/s]
z-zel[µm]
2
Figure 5. Distributions of the pressure and velocity along the axes of the electrode. (a) correspond to V0 10 kV, (b) −12.2 kV, (c) −20 kV. Inthe first column of figures the results of calculation at time t = 4 ns are shown. Curves 1 correspond to hydrostatic pressure p, 2 is the totalpressure ptotal, 3 is the electrostriction pressure pE. In columns 2 and 3 dependences total pressure and velocity u are shown at differentmoments of time. Curves 1—t = 1 ns, 2—2 ns, 3—3 ns, 4—4 ns. zel—coordinate of the edge of the sharp electrode.
Let us estimate the size of the cavitation area. Since the12 kV voltage amplitude corresponds to the maximum negativepressure equal to 12 Mpa, and the 10 kV–10 Mpa, therefore,using threshold pressure of ∼11 MPa, we can estimate the sizeof cavitation region R
(num)cav = 18 µm at V0 = 20 kV. This size
agrees with experimental observations, R(exp)cav = 15–17 µm. It
is important to notice that the electrostrictive negative pressuredepends on the electrode tip radius as pE ∝ E2 ∝ 1/R2
0 , andtherefore 30% uncertainty of the tip radius measurement leadsto a change of threshold pressure value by two times.
Perturbation of liquid density δρ/ρ0 for different timesfor the case V0 = 20 kV was presented in figure 6 and 7. It iseasy to see the density perturbation evolution. This explainsthe presence of the bright spots observed in figure 4. Theposition of the maxima of luminosity in figure 4 coincideswith the position of the maximum density of the liquid infigure 6. Thus, at t = 88 ns, in the experiment the maximumluminance is at position l ≈ 93 µm from the electrode(figure 4), and the corresponding calculated maximum densityat l ≈ 111 µm (figure 6, line 5).
0 20 40 60 80 100 120 140-0.004
-0.002
0.000
0.002
0.004
0.006
0.008
0.010
0.012
0.014
4
1
2
3
δρ/ρ
0
z-zel[µm]
5
Figure 6. Distributions of perturbation density δρ/ρ0 along the axesof the electrode. Curves 1—t = 20 ns, 2—24 ns, 3—28 ns,4—40 ns, 5—88 ns.
4
J. Phys. D: Appl. Phys. 47 (2014) 025502 M Pekker et al
r[um]
z[um
]
0 20 40 60 80 100100
120
140
160
180
200
220
240
260
280
300
320
340
360
6
7
8
9
10
11
12x 10
-3
r[um]
z[um
]0 20 40 60 80 100
100
120
140
160
180
200
220
240
260
280
300
320
340
360
4
4.5
5
5.5
6
6.5
7
7.5
8x 10
-3
r[um]
z[um
]
0 30 60 90 120 150 180 210 2400
30
60
90
120
150
180
210
240
270
300
330
360
390
420
450
1
1.2
1.4
1.6
1.8
2
2.2x 10
-3
t=20ns t=24ns t=88ns
Figure 7. Two-dimensional distribution of perturbation density δρ/ρ0. The black line corresponds to the electrode.
5. Conclusions
It is shown that, in accordance with the theory, rapid pulsedvoltage applied to the needle electrode, creates discontinuitiesin the fluid which can be detected by an optical method. Thereis good agreement between experiment and theory in all areasof measurements.
References
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